This paper completely characterizes the set of Nash equilibria of the Vickrey auction for multiple identical units when buyers have non-increasing marginal valuations and there at least three potential buyers. There are two types of equilibria: In the first class of equilibria there are positive bids below the maximum valuation. In this class, a...

This paper completely characterizes the set of Nash equilibria of the Vickrey auction for multiple identical units when buyers have non-increasing marginal valuations and there at least three potential buyers. There are two types of equilibria: In the first class of equilibria there are positive bids below the maximum valuation. In this class, above a threshold value all bidders bid truthfully on all units. One of the bidders bids at the threshold for any unit for which his valuation is below the threshold; the other bidders bid zero in this range. In the second class of equilibria there are as many bids at or above the maximum valuation as there are units. The allocation of these bids is arbitrary across bidders. All the remaining bids equal zero. With any positive reserve price equilibrium becomes unique: Bidders bid truthfully on all units for which their valuation exceeds the reserve price. ; Vickrey auction, Multi-unit auction, ex-post equilibrium, reserve price, uniqueness Minimize

This paper completely characterizes the set of equilibria of the Vickrey auction for multiple identical units when buyers have non-increasing marginal valuations and there are at least three potential buyers. There are two types of equilibria: In the first class of equilibria there are positive bids below the maximum valuation. In this class, ab...

This paper completely characterizes the set of equilibria of the Vickrey auction for multiple identical units when buyers have non-increasing marginal valuations and there are at least three potential buyers. There are two types of equilibria: In the first class of equilibria there are positive bids below the maximum valuation. In this class, above a threshold value all bidders bid truthfully on all units. One of the bidders bids at the threshold for any unit for which his valuation is below the threshold; the other bidders bid zero in this range. In the second class of equilibria there are as many bids at or above the maximum valuation as there are units. The allocation of these bids is arbitrary across bidders. All the remaining bids equal zero. With any positive reserve price equilibrium becomes unique: Bidders bid truthfully on all units for which their valuation exceeds the reserve price. Minimize

A Traditional Chinese Medicine (TCM) formula is a collection of several herbs. TCM formulae have been used to treat various diseases for several thousand years. However, wide usage of TCM formulae has results in rapid decline of some rare herbs. So it is urgent to find common available replacements for those rare herbs with the similar effects. ...

A Traditional Chinese Medicine (TCM) formula is a collection of several herbs. TCM formulae have been used to treat various diseases for several thousand years. However, wide usage of TCM formulae has results in rapid decline of some rare herbs. So it is urgent to find common available replacements for those rare herbs with the similar effects. In addition, a formula can be simplified by reducing herbs with unchanged effects. Based on this consideration, we propose a method, called “formula pair,” to replace the rare herbs and simplify TCM formulae. We show its reasonableness from a perspective of pathway enrichment analysis. Both the replacements of rare herbs and simplifications of formulae provide new approaches for a new formula discovery. We demonstrate our approach by replacing a rare herb “Forsythia suspensa” in the formula “the seventh of Sang Ju Yin plus/minus herbs (SSJY)” with a common herb “Thunberg Fritillary Bulb” and simplifying two formulae, “the fifth of Du Huo Ji Sheng Tang plus/minus herbs (FDHJST)” and “Fang Feng Tang” (FFT) to a new formula “Fang Feng Du Huo Tang” (FFDHT). Minimize